Questions tagged [complex-manifolds]

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Doubt about proof Proper Mapping Theorem (Remmert)

I am studying the proof of Remmert's Theorem on the book Griffiths & Harris - Principles of Algebraic Geometry, Chapter 3 Section 2 page 395. Theorem (Remmert's Proper Mapping Theorem) Let $U$ and ...
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Regarding the Definition of Harmonic Current on a Compact Complex Manifold

Harmonic Currents of Finite Energy and Laminations, J. E. Fornaess and N. Sibony, Geometric And Functional Analysis, $2005$, Page $965$, Section $2$ "Harmonic Currents". Definition $2.1$ Let ...
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Non-elementary Fuchsian Group and the Invariant Measure for the Corresponding Action on the Limit Set

Let $\Gamma$ be a Fuchsian Group (that means a Discrete Subgroup of $RSL_2(\mathbb{R})$) acting on the Closed Unit Disc $\mathbb{D}$. Let $\Lambda (\Gamma)$ be the Limit Set of $\Gamma$, i.e., it is ...
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Restrictions on coordinate basis of $T_pM$ required for a manifold to admit a Hermitian metric?

I asked this question about relating the Riemannian metric on a manifold $M$ to the Hermitian metric that arises when $M$ is thought of as a complex manifold (i.e. with integrable complex structure). ...
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Growth of Leaves of a Foliation on a Manifold

Let $M$ be a (Real or Complex) Manifold of dimension $m$ and $\mathcal{F}$ a Foliation of dimension $k$, $1 \leq k \leq m-1$, on $M$. I am actually looking for highly recommended references on the ...
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Is the complexification $M^{\mathbb{C}}$ of a complex manifold $M$ fibered over $M$?

Denote by $\mathcal{O}_{\cdot,p}$ the $\mathbb{C}$-algebra of a complex manifold at a given point $p$. Let $M$ be a complex manifold and $M^{\mathbb{C}}$ the complexification of the associated real ...
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How to Prove a Riemann Surface is Hyperbolic by Showing its Volume Grows Exponentially?

We have a Holomorphic Foliation by curves on a Complex Manifold. Therefore, every leaf is a Riemann Surface. Let $R$ be a Riemann Surface. We say that the Riemann Surface $R$ is Hyperbolic if and only ...
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Linearisability of Holomorphic Vector Fields and Poincaré-Dulac Normal Form

Complex Algebraic Foliations, Book by A. Lins Neto and Bruno Scárdua, Chapter 1, Page 36. The last two lines, Don't you think they should have written that the vector field is locally conjugate (...
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How is The Number of Self-Intersections of a Compact Connected Riemann Surface Embedded in a Complex Surface defined?

Complex Algebraic Foliations, Book by A. Lins Neto and Bruno Scárdua, Chapter 3, Page 82. Let $M$ be a Complex Manifold of complex dimension two (i.e., $M$ is a Complex Surface). Let $S$ be a Compact ...
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The Eigenvalues of a Singular One-Dimensional Holomorphic Foliation at a Singular Point

Let $M$ be a Complex Manifold of complex dimension $n$. Let $\mathcal{F}$ be a Singular One-Dimensional Holomorphic Foliation on the complex manifold $M$. Let $p \in M$ be a Singular Point of the ...
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Regarding the Definition of Holomorphic Foliation on a Complex Manifold

Geometry, Dynamics And Topology Of Foliations: A First Course, Book by Bruno Scárdua and Carlos Arnoldo Morales Rojas, Chapter 1, Page 33. Definition 1.14. The definition of Holomorphic Foliation on a ...
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Holomorphic and Harmonic $1-$ Forms on Riemann Surfaces

Let $\mathbb{S}$ be a Riemann Surface. We say that a differential form $\omega$, on the Riemann Surface $\mathbb{S}$, is a Harmonic $1-$ Form if it and its conjugate $\omega^\star$ are both closed, i....
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Easier proof that the Grassmannian is a complex manifold

$G_r(\mathbb C^3,2)$ is the topological space of 2-dimensional complex linear subspaces of $\mathbb C^3$. Prove that $G_r(\mathbb C^3,2)$ is a complex manifold. I have a solution to this problem, but ...
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Questions about algebraic analysis: prerequesites, references and state of the field [closed]

I'm interested in studying algebraic analysis, specifically the area described in here https://en.wikipedia.org/wiki/Algebraic_analysis and also the theory of D-modules, as I find the idea of "...
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Holomorphic function on a connected compact Riemann surface is constant

I was trying to solve the following exercise. I wanted to check if my solution was correct/rigorous enough, and ask a question at the end. (The general direction is given in here: holomorphic map ...
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Existence of coframe for Hermitian metric on complex manifold?

I am reading page 28 of the 1994 version of Principles of Algebraic Geometry by Griffith. Let $M$ be a complex manifold of dimension n, Griffith defined a Hermitian metric to be a positive definite ...
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Connected components of the isotropic Grassmannian

Let $W$ be a $2n$-dimensional complex vector space endowed with a non-degenerate, symmetric, bilinear form $Q$. We choose Euclidean coordinates on $W$ such that $Q$ is represented by symmetric ...
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Conditions on $X,\Omega$ such that $g(K)$ is a Stein compactum in $X$ $\forall g:\Omega\to X$ holomorphic and $\forall K\Subset\Omega$.

Let $\Omega\subset\Bbb C^n$ open bounded and $X$ complex manifold. I am searching for some condition on $X,\Omega$ such that $g(K)$ is a Stein compactum in $X$ for every $g:\Omega\to X$ holomorphic ...
Let $Y$ be a closed complex surface, $L\to Y$ be a holomorphic line bundle, $\sigma:Y\to L$ a holomorphic section, and $B\subset Y$ the zero set of $\sigma$. If the first Chern class $c_1(L)$ of $L$ ...
On the complexification of a holomorphic bundle $E$
Let $E\to M$ be a holomorphic vector bundle over a complex manifold. Considering it as a real vector bundle one can complexify $E$ to $E^{\mathbb{C}}$. The bundle $E^{\mathbb{C}}$ is again holomorphic ...